Department of Mathematics, University of Auckland, Auckland, New Zealand.
J Math Neurosci. 2011 Sep 23;1(1):9. doi: 10.1186/2190-8567-1-9.
A major obstacle in the analysis of many physiological models is the issue of model simplification. Various methods have been used for simplifying such models, with one common technique being to eliminate certain 'fast' variables using a quasi-steady-state assumption. In this article, we show when such a physiological model reduction technique in a slow-fast system is mathematically justified. We provide counterexamples showing that this technique can give erroneous results near the onset of oscillatory behaviour which is, practically, the region of most importance in a model. In addition, we show that the singular limit of the first Lyapunov coefficient of a Hopf bifurcation in a slow-fast system is, in general, not equal to the first Lyapunov coefficient of the Hopf bifurcation in the corresponding layer problem, a seemingly counterintuitive result. Consequently, one cannot deduce, in general, the criticality of a Hopf bifurcation in a slow-fast system from the lower-dimensional layer problem.
在分析许多生理模型时，主要的障碍是模型简化问题。已经使用了各种方法来简化这些模型，一种常见的技术是使用准稳态假设来消除某些“快速”变量。在本文中，我们展示了在慢快系统中使用这种生理模型简化技术在数学上是合理的。我们提供了反例，表明该技术在接近振荡行为开始时会给出错误的结果，而在模型中，这实际上是最重要的区域。此外，我们还表明，在慢快系统中，Hopf 分岔的第一 Lyapunov 系数的奇异极限通常不等于相应层问题中 Hopf 分岔的第一 Lyapunov 系数，这是一个看似违反直觉的结果。因此，一般来说，不能从低维层问题推断慢快系统中 Hopf 分岔的临界点。
